Question

A cylinder of mass m is suspended through two strings wrapped around it as shwon in figure . Find a. the tension T in the string and b. the speed o the cylinder as it falls through a distance h.

Answer 1

The portion of th strings between the cweiling and the cylinder is at rest. Hence the oints o the cuylinder where the strings leave it are at rest. The cylinder is thus rolling withuot sliping on the strings. Suppose the centre of the cylinder falls with n acceleration a. The ngular acceleration of he cylenere about its axis s `alpha=a/R`, as the cylinder does not slip ovr teh strings.
The equation of motion for the centre of mass of the cylinder is
mg-2T=ma`.............i
and for the motion about the centre of mass it as
`2Tr=(1/2mr^2alpha)=1/2mra`
or `2T=1/2ma`............ii
From i and ii
`a=2/3g and T=(mg)/6`
As the centre of the cylinder starts moving from rest, teh velocity after it hs fallen through a distance h is given by
`v^2=2(2/3g)h`
or `v=(sqrt(4gh))/3`